Properties

Label 3450.2.a.g.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} -5.00000 q^{19} -3.00000 q^{21} +1.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -3.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -9.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{36} +5.00000 q^{38} -3.00000 q^{39} -5.00000 q^{41} +3.00000 q^{42} -7.00000 q^{43} -1.00000 q^{44} +1.00000 q^{46} +2.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +3.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -3.00000 q^{56} +5.00000 q^{57} +9.00000 q^{58} -12.0000 q^{59} +4.00000 q^{61} +2.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +8.00000 q^{67} +1.00000 q^{69} -10.0000 q^{71} -1.00000 q^{72} -5.00000 q^{73} -5.00000 q^{76} -3.00000 q^{77} +3.00000 q^{78} +9.00000 q^{79} +1.00000 q^{81} +5.00000 q^{82} -11.0000 q^{83} -3.00000 q^{84} +7.00000 q^{86} +9.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} +9.00000 q^{91} -1.00000 q^{92} +2.00000 q^{93} -2.00000 q^{94} +1.00000 q^{96} +14.0000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 1.00000 0.213201
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 5.00000 0.811107
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 3.00000 0.462910
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 5.00000 0.662266
\(58\) 9.00000 1.18176
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 2.00000 0.254000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.00000 −0.585206 −0.292603 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) −3.00000 −0.341882
\(78\) 3.00000 0.339683
\(79\) 9.00000 1.01258 0.506290 0.862364i \(-0.331017\pi\)
0.506290 + 0.862364i \(0.331017\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.00000 0.552158
\(83\) −11.0000 −1.20741 −0.603703 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) 7.00000 0.754829
\(87\) 9.00000 0.964901
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 9.00000 0.943456
\(92\) −1.00000 −0.104257
\(93\) 2.00000 0.207390
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −2.00000 −0.202031
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −5.00000 −0.468293
\(115\) 0 0
\(116\) −9.00000 −0.835629
\(117\) 3.00000 0.277350
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −4.00000 −0.362143
\(123\) 5.00000 0.450835
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.00000 0.616316
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 1.00000 0.0870388
\(133\) −15.0000 −1.30066
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 10.0000 0.839181
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 5.00000 0.413803
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 5.00000 0.405554
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) −3.00000 −0.240192
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −9.00000 −0.716002
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) −1.00000 −0.0785674
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) 11.0000 0.853766
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 3.00000 0.231455
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) −7.00000 −0.533745
\(173\) −3.00000 −0.228086 −0.114043 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) −10.0000 −0.749532
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) −9.00000 −0.667124
\(183\) −4.00000 −0.295689
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) 2.00000 0.145865
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −11.0000 −0.783718 −0.391859 0.920025i \(-0.628168\pi\)
−0.391859 + 0.920025i \(0.628168\pi\)
\(198\) 1.00000 0.0710669
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 6.00000 0.422159
\(203\) −27.0000 −1.89503
\(204\) 0 0
\(205\) 0 0
\(206\) 11.0000 0.766406
\(207\) −1.00000 −0.0695048
\(208\) 3.00000 0.208013
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 2.00000 0.137361
\(213\) 10.0000 0.685189
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −6.00000 −0.407307
\(218\) 16.0000 1.08366
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 5.00000 0.331133
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 9.00000 0.590879
\(233\) 5.00000 0.327561 0.163780 0.986497i \(-0.447631\pi\)
0.163780 + 0.986497i \(0.447631\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −9.00000 −0.584613
\(238\) 0 0
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 10.0000 0.642824
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) −5.00000 −0.318788
\(247\) −15.0000 −0.954427
\(248\) 2.00000 0.127000
\(249\) 11.0000 0.697097
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 3.00000 0.188982
\(253\) 1.00000 0.0628695
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −7.00000 −0.435801
\(259\) 0 0
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) −2.00000 −0.123560
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 15.0000 0.919709
\(267\) −10.0000 −0.611990
\(268\) 8.00000 0.488678
\(269\) −17.0000 −1.03651 −0.518254 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) −9.00000 −0.544705
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 21.0000 1.26177 0.630884 0.775877i \(-0.282692\pi\)
0.630884 + 0.775877i \(0.282692\pi\)
\(278\) −12.0000 −0.719712
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 2.00000 0.119098
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) −15.0000 −0.885422
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) −5.00000 −0.292603
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) −22.0000 −1.27443
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −21.0000 −1.21042
\(302\) 2.00000 0.115087
\(303\) 6.00000 0.344691
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −3.00000 −0.170941
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 3.00000 0.169842
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 9.00000 0.506290
\(317\) −3.00000 −0.168497 −0.0842484 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 2.00000 0.112154
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 3.00000 0.167183
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) 16.0000 0.884802
\(328\) 5.00000 0.276079
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) −11.0000 −0.603703
\(333\) 0 0
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 4.00000 0.217571
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 5.00000 0.270369
\(343\) −15.0000 −0.809924
\(344\) 7.00000 0.377415
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 9.00000 0.482451
\(349\) 11.0000 0.588817 0.294408 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 1.00000 0.0533002
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −11.0000 −0.580558 −0.290279 0.956942i \(-0.593748\pi\)
−0.290279 + 0.956942i \(0.593748\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 12.0000 0.630706
\(363\) 10.0000 0.524864
\(364\) 9.00000 0.471728
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 11.0000 0.574195 0.287098 0.957901i \(-0.407310\pi\)
0.287098 + 0.957901i \(0.407310\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 2.00000 0.103695
\(373\) −12.0000 −0.621336 −0.310668 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) −27.0000 −1.39057
\(378\) 3.00000 0.154303
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 5.00000 0.255822
\(383\) −1.00000 −0.0510976 −0.0255488 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −7.00000 −0.355830
\(388\) 14.0000 0.710742
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.00000 −0.101015
\(393\) −2.00000 −0.100887
\(394\) 11.0000 0.554172
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 3.00000 0.150376
\(399\) 15.0000 0.750939
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 8.00000 0.399004
\(403\) −6.00000 −0.298881
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 27.0000 1.33999
\(407\) 0 0
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) −11.0000 −0.541931
\(413\) −36.0000 −1.77144
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) −12.0000 −0.587643
\(418\) −5.00000 −0.244558
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −10.0000 −0.486792
\(423\) 2.00000 0.0972433
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −10.0000 −0.484502
\(427\) 12.0000 0.580721
\(428\) 4.00000 0.193347
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 5.00000 0.239182
\(438\) −5.00000 −0.238909
\(439\) −36.0000 −1.71819 −0.859093 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) −22.0000 −1.04056
\(448\) 3.00000 0.141737
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 2.00000 0.0940721
\(453\) 2.00000 0.0939682
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −5.00000 −0.234146
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) 35.0000 1.63011 0.815056 0.579382i \(-0.196706\pi\)
0.815056 + 0.579382i \(0.196706\pi\)
\(462\) −3.00000 −0.139573
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) −5.00000 −0.231621
\(467\) −17.0000 −0.786666 −0.393333 0.919396i \(-0.628678\pi\)
−0.393333 + 0.919396i \(0.628678\pi\)
\(468\) 3.00000 0.138675
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 12.0000 0.552345
\(473\) 7.00000 0.321860
\(474\) 9.00000 0.413384
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 26.0000 1.18921
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) 3.00000 0.136505
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −4.00000 −0.181071
\(489\) 22.0000 0.994874
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 5.00000 0.225417
\(493\) 0 0
\(494\) 15.0000 0.674882
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −30.0000 −1.34568
\(498\) −11.0000 −0.492922
\(499\) 30.0000 1.34298 0.671492 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) 0 0
\(501\) 2.00000 0.0893534
\(502\) −24.0000 −1.07117
\(503\) 33.0000 1.47140 0.735699 0.677309i \(-0.236854\pi\)
0.735699 + 0.677309i \(0.236854\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −1.00000 −0.0444554
\(507\) 4.00000 0.177646
\(508\) 12.0000 0.532414
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −15.0000 −0.663561
\(512\) −1.00000 −0.0441942
\(513\) 5.00000 0.220755
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 7.00000 0.308158
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 9.00000 0.393919
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −15.0000 −0.650332
\(533\) −15.0000 −0.649722
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 6.00000 0.258919
\(538\) 17.0000 0.732922
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 2.00000 0.0859074
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) 0 0
\(546\) 9.00000 0.385164
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 45.0000 1.91706
\(552\) −1.00000 −0.0425628
\(553\) 27.0000 1.14816
\(554\) −21.0000 −0.892205
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 36.0000 1.52537 0.762684 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 2.00000 0.0846668
\(559\) −21.0000 −0.888205
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −31.0000 −1.30649 −0.653247 0.757145i \(-0.726594\pi\)
−0.653247 + 0.757145i \(0.726594\pi\)
\(564\) −2.00000 −0.0842152
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 3.00000 0.125988
\(568\) 10.0000 0.419591
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) −3.00000 −0.125436
\(573\) 5.00000 0.208878
\(574\) 15.0000 0.626088
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 47.0000 1.95664 0.978318 0.207109i \(-0.0664056\pi\)
0.978318 + 0.207109i \(0.0664056\pi\)
\(578\) 17.0000 0.707107
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −33.0000 −1.36907
\(582\) 14.0000 0.580319
\(583\) −2.00000 −0.0828315
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) 16.0000 0.660954
\(587\) −44.0000 −1.81607 −0.908037 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 11.0000 0.452480
\(592\) 0 0
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 3.00000 0.122782
\(598\) 3.00000 0.122679
\(599\) 38.0000 1.55264 0.776319 0.630340i \(-0.217085\pi\)
0.776319 + 0.630340i \(0.217085\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 21.0000 0.855896
\(603\) 8.00000 0.325785
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 5.00000 0.202777
\(609\) 27.0000 1.09410
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −11.0000 −0.442485
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −16.0000 −0.641542
\(623\) 30.0000 1.20192
\(624\) −3.00000 −0.120096
\(625\) 0 0
\(626\) 16.0000 0.639489
\(627\) −5.00000 −0.199681
\(628\) 6.00000 0.239426
\(629\) 0 0
\(630\) 0 0
\(631\) −37.0000 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) −9.00000 −0.358001
\(633\) −10.0000 −0.397464
\(634\) 3.00000 0.119145
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 6.00000 0.237729
\(638\) −9.00000 −0.356313
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) 4.00000 0.157867
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) 0 0
\(647\) −30.0000 −1.17942 −0.589711 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) −22.0000 −0.861586
\(653\) 23.0000 0.900060 0.450030 0.893014i \(-0.351413\pi\)
0.450030 + 0.893014i \(0.351413\pi\)
\(654\) −16.0000 −0.625650
\(655\) 0 0
\(656\) −5.00000 −0.195217
\(657\) −5.00000 −0.195069
\(658\) −6.00000 −0.233904
\(659\) −3.00000 −0.116863 −0.0584317 0.998291i \(-0.518610\pi\)
−0.0584317 + 0.998291i \(0.518610\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 26.0000 1.01052
\(663\) 0 0
\(664\) 11.0000 0.426883
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) −2.00000 −0.0773823
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 3.00000 0.115728
\(673\) −11.0000 −0.424019 −0.212009 0.977268i \(-0.568001\pi\)
−0.212009 + 0.977268i \(0.568001\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −24.0000 −0.922395 −0.461197 0.887298i \(-0.652580\pi\)
−0.461197 + 0.887298i \(0.652580\pi\)
\(678\) 2.00000 0.0768095
\(679\) 42.0000 1.61181
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) −2.00000 −0.0765840
\(683\) 46.0000 1.76014 0.880071 0.474843i \(-0.157495\pi\)
0.880071 + 0.474843i \(0.157495\pi\)
\(684\) −5.00000 −0.191180
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) 14.0000 0.534133
\(688\) −7.00000 −0.266872
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −3.00000 −0.114043
\(693\) −3.00000 −0.113961
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 0 0
\(698\) −11.0000 −0.416356
\(699\) −5.00000 −0.189117
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 3.00000 0.113228
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −3.00000 −0.112906
\(707\) −18.0000 −0.676960
\(708\) 12.0000 0.450988
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 9.00000 0.337526
\(712\) −10.0000 −0.374766
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 26.0000 0.970988
\(718\) 11.0000 0.410516
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) −33.0000 −1.22898
\(722\) −6.00000 −0.223297
\(723\) −14.0000 −0.520666
\(724\) −12.0000 −0.445976
\(725\) 0 0
\(726\) −10.0000 −0.371135
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −9.00000 −0.333562
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −4.00000 −0.147844
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −11.0000 −0.406017
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −8.00000 −0.294684
\(738\) 5.00000 0.184053
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) 15.0000 0.551039
\(742\) −6.00000 −0.220267
\(743\) 39.0000 1.43077 0.715386 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) −11.0000 −0.402469
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 2.00000 0.0729325
\(753\) −24.0000 −0.874609
\(754\) 27.0000 0.983282
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 20.0000 0.726433
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) 25.0000 0.906249 0.453125 0.891447i \(-0.350309\pi\)
0.453125 + 0.891447i \(0.350309\pi\)
\(762\) 12.0000 0.434714
\(763\) −48.0000 −1.73772
\(764\) −5.00000 −0.180894
\(765\) 0 0
\(766\) 1.00000 0.0361315
\(767\) −36.0000 −1.29988
\(768\) −1.00000 −0.0360844
\(769\) 24.0000 0.865462 0.432731 0.901523i \(-0.357550\pi\)
0.432731 + 0.901523i \(0.357550\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) 40.0000 1.43870 0.719350 0.694648i \(-0.244440\pi\)
0.719350 + 0.694648i \(0.244440\pi\)
\(774\) 7.00000 0.251610
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 25.0000 0.895718
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) 27.0000 0.962446 0.481223 0.876598i \(-0.340193\pi\)
0.481223 + 0.876598i \(0.340193\pi\)
\(788\) −11.0000 −0.391859
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 1.00000 0.0355335
\(793\) 12.0000 0.426132
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −3.00000 −0.106332
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) −15.0000 −0.530994
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 12.0000 0.423735
\(803\) 5.00000 0.176446
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) 17.0000 0.598428
\(808\) 6.00000 0.211079
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) −27.0000 −0.947514
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 35.0000 1.22449
\(818\) −25.0000 −0.874105
\(819\) 9.00000 0.314485
\(820\) 0 0
\(821\) 29.0000 1.01211 0.506053 0.862502i \(-0.331104\pi\)
0.506053 + 0.862502i \(0.331104\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 11.0000 0.383203
\(825\) 0 0
\(826\) 36.0000 1.25260
\(827\) −37.0000 −1.28662 −0.643308 0.765607i \(-0.722439\pi\)
−0.643308 + 0.765607i \(0.722439\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 27.0000 0.937749 0.468874 0.883265i \(-0.344660\pi\)
0.468874 + 0.883265i \(0.344660\pi\)
\(830\) 0 0
\(831\) −21.0000 −0.728482
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 5.00000 0.172929
\(837\) 2.00000 0.0691301
\(838\) −9.00000 −0.310900
\(839\) 3.00000 0.103572 0.0517858 0.998658i \(-0.483509\pi\)
0.0517858 + 0.998658i \(0.483509\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 26.0000 0.896019
\(843\) −6.00000 −0.206651
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) −2.00000 −0.0687614
\(847\) −30.0000 −1.03081
\(848\) 2.00000 0.0686803
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 10.0000 0.342594
\(853\) 31.0000 1.06142 0.530710 0.847554i \(-0.321925\pi\)
0.530710 + 0.847554i \(0.321925\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −3.00000 −0.102418
\(859\) 16.0000 0.545913 0.272956 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) 0 0
\(861\) 15.0000 0.511199
\(862\) −8.00000 −0.272481
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −28.0000 −0.951479
\(867\) 17.0000 0.577350
\(868\) −6.00000 −0.203653
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 16.0000 0.541828
\(873\) 14.0000 0.473828
\(874\) −5.00000 −0.169128
\(875\) 0 0
\(876\) 5.00000 0.168934
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 36.0000 1.21494
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) 4.00000 0.134307 0.0671534 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) 0 0
\(889\) 36.0000 1.20740
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 2.00000 0.0669650
\(893\) −10.0000 −0.334637
\(894\) 22.0000 0.735790
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 3.00000 0.100167
\(898\) −22.0000 −0.734150
\(899\) 18.0000 0.600334
\(900\) 0 0
\(901\) 0 0
\(902\) −5.00000 −0.166482
\(903\) 21.0000 0.698836
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) 31.0000 1.02934 0.514669 0.857389i \(-0.327915\pi\)
0.514669 + 0.857389i \(0.327915\pi\)
\(908\) −20.0000 −0.663723
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 5.00000 0.165657 0.0828287 0.996564i \(-0.473605\pi\)
0.0828287 + 0.996564i \(0.473605\pi\)
\(912\) 5.00000 0.165567
\(913\) 11.0000 0.364047
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −35.0000 −1.15266
\(923\) −30.0000 −0.987462
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) −20.0000 −0.657241
\(927\) −11.0000 −0.361287
\(928\) 9.00000 0.295439
\(929\) 51.0000 1.67326 0.836628 0.547772i \(-0.184524\pi\)
0.836628 + 0.547772i \(0.184524\pi\)
\(930\) 0 0
\(931\) −10.0000 −0.327737
\(932\) 5.00000 0.163780
\(933\) −16.0000 −0.523816
\(934\) 17.0000 0.556257
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −24.0000 −0.783628
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) −26.0000 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 6.00000 0.195491
\(943\) 5.00000 0.162822
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −7.00000 −0.227590
\(947\) 38.0000 1.23483 0.617417 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) −9.00000 −0.292306
\(949\) −15.0000 −0.486921
\(950\) 0 0
\(951\) 3.00000 0.0972817
\(952\) 0 0
\(953\) 16.0000 0.518291 0.259145 0.965838i \(-0.416559\pi\)
0.259145 + 0.965838i \(0.416559\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −26.0000 −0.840900
\(957\) −9.00000 −0.290929
\(958\) 21.0000 0.678479
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) −3.00000 −0.0965234
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 36.0000 1.15411
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) −22.0000 −0.703482
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 12.0000 0.382935
\(983\) −3.00000 −0.0956851 −0.0478426 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) −5.00000 −0.159394
\(985\) 0 0
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) −15.0000 −0.477214
\(989\) 7.00000 0.222587
\(990\) 0 0
\(991\) 54.0000 1.71537 0.857683 0.514178i \(-0.171903\pi\)
0.857683 + 0.514178i \(0.171903\pi\)
\(992\) 2.00000 0.0635001
\(993\) 26.0000 0.825085
\(994\) 30.0000 0.951542
\(995\) 0 0
\(996\) 11.0000 0.348548
\(997\) 25.0000 0.791758 0.395879 0.918303i \(-0.370440\pi\)
0.395879 + 0.918303i \(0.370440\pi\)
\(998\) −30.0000 −0.949633
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3450.2.a.g.1.1 1
5.2 odd 4 3450.2.d.o.2899.1 2
5.3 odd 4 3450.2.d.o.2899.2 2
5.4 even 2 3450.2.a.v.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.g.1.1 1 1.1 even 1 trivial
3450.2.a.v.1.1 yes 1 5.4 even 2
3450.2.d.o.2899.1 2 5.2 odd 4
3450.2.d.o.2899.2 2 5.3 odd 4